Line formation in moving media: Asymptotic expansions of some special functions

Line formation in the spectrum of a moving medium with a spherical geometry is considered. In the Sobolev approximation there are some special functions that determine the source function and the force of radiation pressure in the line. The most important case is that of a small dimensionless velocity gradient (i.e., a large dimensionless Sobolev length τ) and a small ratio β of the opacity in the continuum to the opacity in the line. Until now there has been no detailed analytical information about the asymptotic behavior of these functions. For the case of a Doppler profile of the absorption coefficient, we clarify the nontrivial structure of their total asymptotic expansions for τ » 1, β « 1, and arbitrary Βτ. We give an algorithm for obtaining all the coefficients of these expansions and give explicit expressions for the first few coefficients. We also compare the asymptotic expansions with the numerical calculations of these functions available in the literature. We also briefly consider the case of a power-law decrease in the absorption coefficient in the line wing (and, in more detail, the case of Lorentz wings of the Voigt profile).